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Handbook of Memristor Networks pp 89–157Cite as

Everything You Wish to Know About Memristors but Are Afraid to Ask

Abstract

This paper classifies all memristors into three classes called Ideal, Generic, or Extended memristors. A subclass of Generic memristors is related to Ideal memristors via a one-to-one mathematical transformation, and is hence called Ideal Generic memristors. The concept of non-volatile memories is defined and clarified with illustrations. Several fundamental new concepts, including Continuum-memory memristor, POP (acronym for Power-Off Plot), DC V-I Plot, and Quasi DC V-I Plot, are rigorously defined and clarified with colorful illustrations. Among many colorful pictures the shoelace DC V-I Plot stands out as both stunning and illustrative. Even more impressive is that this bizarre shoelace plot has an exact analytical representation via 2 explicit functions of the state variable, derived by a novel parametric approach invented by the author.

Keywords

  • Memristor
  • Continuum-memory memristor
  • POP
  • Power-Off plot
  • DC V-I plot
  • Quasi DC V-I plot
  • Shoelace V-I plot
  • Parametric approach
  • Graphical composition
  • Piecewise-Linear function (PWL)

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Notes

  1. 1.

    Our axiomatic definition requires a pinched hysteresis loop to be measured not only for one input signal, but for all possible periodic input signals with zero mean. In practice, only a finite number of measurements could be made. Our definition did not require the same pinched hysteresis loop to be measured whenever the same input signal is applied, because for non-volatile memristors, the pinched hysteresis loop depends not only on the input waveform i(t) or v(t), but also on the initial conditions of the relevant state variables, such as Fig. 42 of [5], where two different pinched hysteresis loops are measured for the same input current source i(t) = 10 sinωt, but different initial states x(0) = −6.3 and x(0) = 6.3, respectively.

  2. 2.

    Readers puzzled by the dramatic difference between the 2 pinched hysteresis loops in Fig. 4b, d calculated from the same memristor constitutive relation \( q = \hat{q}(\varphi ) \), and the same input v(t) = 1.2 sint are referred to the exact calculations and graphical illustrations in Figs. 28 and  27 of [5], respectively.

  3. 3.

    Observe the two PWL Eqs. (8) and (9) give rise to two different PWL curves in Figs. 4a and 5a, respectively. Unlike Fig. 4a, the PWL curve in Fig. 5a is not odd symmetric due to the presence of a constant term in (9).

  4. 4.

    Although one can choose any piecewise-differentiable function \( x = \hat{x}(\varphi ) \), it is rare that its inverse function \( \varphi = \hat{x}^{ - 1} (x) \) has an analytical equation. We opted for a PWL function because not only it has an explicit equation, as shown in Table 6 of Appendix , its inverse function is also a PWL function, and hence will also have an explicit equation.

  5. 5.

    Recall the 1:1 function we used to derive the memristor sibling in Fig. 10 is defined by \( x = \hat{x}\,(\varphi ) = 2.125\,\varphi - 1.875\left| \varphi \right| \).

  6. 6.

    The same algorithm applies for voltage-controlled memristors .

  7. 7.

    A current-controlled (resp., voltage-controlled) memristors is passive if, and only if, its memristence R(x,i) ≥ 0 (resp. memductance G(x, v) ≥ 0). A memristor is active if, and only if, it is not passive.

  8. 8.

    Fig. 19d illustrates why the Coincident Zero-Crossing signature is more general than that of a pinched hysteresis loop fingerprint; namely, since both i(t) and v(t) waveforms are not periodic, their associated Lissajous figure is not a closed loop, but an unending loci, which if left to continue printing unstopped would eventually lead to a uniformly blue color inside each lobe. But remarkably, both blue lobes would share a common pinched point at the origin at all times, thereby confirming the nonlinear device defined in Fig. 19a is a memristor.

  9. 9.

    The phase shift between two periodic waveforms i(t) and v(t) of the same frequency is the distance in time measured between their two closest zero crossings.

  10. 10.

    The adjective asymptotically is used in a mathematical sense meaning that x(t) will not arrive at x(t)  = x(Q) at a finite time.

  11. 11.

    To simplify arithmetic, we pick G0 = 1. In practice, G0 is a scaling constant chosen to fit the intrinsic memductance scale of the memristor.

  12. 12.

    An equilibrium point x(Q) of a differential equation \( \dot{\varvec{x}} = \varvec{f}(\varvec{x}) \) is said to be asymptotically stable if a small ball placed initially at x(Q) will always return to x(Q) when it is displaced by an arbitrarily small perturbation of arbitrarily short duration by following the direction of motion indicated by the arrowhead along the dynamic route where the perturbed state \( \hat{x} = x(Q) + \Delta x \) is located. If the perturbed state \( \hat{x} \) did not return to x(Q), but remains motionless after the perturbation \( \Delta x \) became zero, then the equilibrium point is said to be stable.

  13. 13.

    For a more comprehensive theory on non-volatile memories, we can generalize our definition of a Continuum Memory Memristor to allow its POP to contain only one, or more, contiguous intervals a  ≤  x  ≤  b on the x-axis, such as the 2 intervals [−2, −1] and [1, 2] in Fig. 35 of [5].

  14. 14.

    We choose the capital letters V and I, instead of the conventional lower case letters v and i to distinguish them from the H F v-i curve (acronym for high-frequency v-i curve) exhibited by all Extended memristors when connected to high-frequency periodic signals.

  15. 15.

    Commercial simulators, such as various versions of SPICE , are even less reliable because the numerical algorithm it uses is incapable of solving complicated nonlinear equations.

  16. 16.

    More accurate solutions can be found by using the graphically derived solutions as initial guess for numerical softwares.

  17. 17.

    The range 300 < XQ < 3000.0008 in Fig. 24 was chosen upon inspection of Fig. 25.

  18. 18.

    In a future paper, we will design an oscillator by connecting a 7-volt battery (with the positive terminal connected to ground) in series with a positive inductor and the memristor defined by (16a–c). This battery will give rise to an unstable equilibrium point located at (−7, −63), thereby spawning a stable limit cycle via a super-critical Hopf Bifurcation mechanism [25].

  19. 19.

    It is possible, however, to design an elaborate experimental set-up to observe the unstable green branch in Fig. 30.

  20. 20.

    Any state equation \( \frac{dx}{dt} = g(x,v) \) where \( g\,(x,0) = 0 \), −∞ < x < ∞ has a continuum memory consisting of all points of the x-axis.

  21. 21.

    Equation (51a) defines a passive memristor because its instantaneous power \( p(t) = i(t)v(t) = x^{2} (t)v^{2} (t) \ge 0 \) for any v(t) and for all times t.

  22. 22.

    It is possible to build a nullator using an op-amp. Indeed, if one connects a resistor from the negative op-amp input terminal of an op-amp to its output terminal, then 2 op-amp input terminals becomes a virtual short circuit, where v = 0 and i  = 0!

  23. 23.

    An ordinary differential equation \( \frac{{d\varvec{x}}}{dt} = {\text{f}}(\varvec{x},t) \) is said to be non-autonomous when the time variable t appear explicitly in the equation, such as the case when the memristor is driven by non-constant voltage source.

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Acknowledgements

The author wishes to thank Prof. Hyongsuk Kim, Zubaer Ibna Mannan, and Cheol Choi for their wonderful assistance in the production of this paper. He would also like to thank Dr. R. Stanley Williams from hp for detecting several errors. The author would like to acknowledge financial support from the USA Air force office of Scientific Research under Grant number FA9550-13-1-0136 and from the European Commission Marie Curie Fellowship, and the EU COST Action IC 1401.

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Authors and Affiliations

  1. Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA, 94530, USA

    Leon Chua

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Editors and Affiliations

  1. Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA, USA

    Prof. Leon Chua

  2. Democritus University of Thrace, Xanthi, Greece

    Dr. Georgios Ch. Sirakoulis

  3. Department of Computer Science and Creative Technologies, University of the West of England, Bristol, UK

    Prof. Andrew Adamatzky

Appendix

Appendix

Since nonlinear algebraic, or differential equations in general has no analytical solutions, they are usually solved by numerical methods, via standard softwares, or circuit simulators , such as SPICE . Unfortunately, numerical softwares are not foolproof, and cannot find all solutions if the equation has more than one solutions. Piecewise-linear (PWL) methods are the best tools in such situations. In order to apply PWL methods, it is often desirable to represent a PWL curve by a PWL equation whose only nonlinearities are the absolute-value function \( y = \left| x \right| \), and the signum function sgn (x), defined in Table 7.

The good news is that unlike other nonlinear basis functions, the coefficients associated with the PWL formula presented in the following Table 6, all coefficients needed to specify any continuous PWL function can be obtained by inspection of the PWL curve! Simply label the the segment number consecutively from left to right, as segment 0, 1, 2,…, n, for an (n + 1)-segment PWL curve, with corresponding slope m0, m1,…, mn. Label the x-coordinate of each corresponding breakpoint as X1, X2,…, Xn. Any continuous PWL curve has a unique PWL formula (shown in Table 6), with 2 coefficients a0, a1; n coefficients X1, X2,…, Xn, and n coefficients b1, b2,…,bn.

The coefficient Xj is equal to the x-coordinate of breakpoint j. The coefficient a1 is equal to half the sum of the slope m0 and mn of the leftmost segment 0 and the rightmost segment n, respectively.

The coefficient bj is equal to half the difference between the slope mj and m(j-1) of segment j and segment ( j−1), respectively. The coefficient a0 is chosen such that y is equal to the vertical intercept f(0) of the PWL curve when x = 0 (see Table 6).

All of these coefficients can be reconstructed from the following mnemonic rule: Half Sum Half Difference then Null.

Table 6 Half sum half difference then null PWL formula
Full size table
Table 7 Graph of the absolute-value function of x, | x |, and the signum function of x, sgn (x). They are the basic functions (building blocks) of all PWL functions
Full size table

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Chua, L. (2019). Everything You Wish to Know About Memristors but Are Afraid to Ask. In: Chua, L., Sirakoulis, G., Adamatzky, A. (eds) Handbook of Memristor Networks. Springer, Cham. https://doi.org/10.1007/978-3-319-76375-0_3

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